Device for analysing a wavefront with enhanced resolution

ABSTRACT

A wavefront analysing device, of the Hartmann or Shack-Hartmann type, comprises in particular a set of sampling elements arranged in an analysis plane, and forming as many micro-lenses for sampling the incident wavefront, and a diffraction plane wherein are analysed the Airy discs of the different micro-lenses illuminated by the incident wavefront. The shape of each micro-lens is such that the associated diffraction figure has in the diffraction plane one or several preferential axe(s), and the microlenses are oriented in the analysis plane such that the preferential axe(s) of the diffraction figure of a micro-lens are offset relative to the preferential axes of the diffraction figures of neighbouring micro-lenses, thereby enabling to limit the overlapping of the diffraction figures.

This invention concerns the field of wavefront analysis using Hartman orShack-Hartmann type methods.

Historically, the Hartmann method for analysing the wavefront introducesthe concept of light sampling by a set of sampling elements with knowncharacteristics and distribution, elements which are, originally,apertures in an opaque screen. The analysis of the light diffracted bythe device enables to trace the shape of the surface of the startingwave, by knowing the positions of the spots on a pre-positioned screenin a plane called thereafter the diffraction plane.

Due to its design, the method needs, however, consequent light fluxesand Shack introduces, in particular to remedy this type of problem, themicro-lenses as light sampling elements.

Improvements in micro-optics, enabling to realise micro-lenses with verydiverse characteristics, render this method particularly flexible: itfinds therefore numerous applications in optical metrology.

Micro-lenses are sized conventionally with large focal lengths withrespect to their dimensions, i.e. they are very slightly open, whichexhibits numerous advantages: high sensitivity to variations of thelocal phase, spot sizes enabling to optimise calculation of theirpositions in particular when the screen is a CCD sensor, minimisingaberrations introduced by these elements.

The micro-lenses thus sized operate most often in diffraction limit:each focussing spot corresponds to a diffraction figure which depends onthe characteristics of the corresponding micro-lens, in particular itsshape. A diffraction figure may extend sufficiently and overlap theneighbouring figure(s) in the diffraction plane, which introduces anerror when calculating the position of the spots and hence the wavesurface. In the case of coherent sources, this overlapping isaccompanied by interference phenomena which are even more detrimental tothe accuracy of measurement.

In order to enhance the resolution of the wavefront analysis device, ithas been sought to increase the number of micro-lenses, and hence toreduce the size thereof, which translates by an increase in size of thediffraction figure and incurs overlapping risks of said figures, whichmay lead to an adverse effect.

One of the solutions to this problem consists in modifying thetransmission of the micro-lenses used in order to obtain betterlocalised diffraction figures, i.e. less spread and therefore lessliable to overlap one another. It is the type of processing recommendedfor example in the international patent application WO 01/04591 A1, bymanufacturing and using an apodisation mask at each sub-pupil. But therealisation of this type of mask, such as centring them at each of thesub-pupils, raises significant technological difficulties.

The solution presented here consists in modifying and optimising thegeometrical arrangement of the sub-pupils used in the Shack-Hartman typeanalyser, for example, in order to limit the overlapping of thediffraction figures in the diffraction plane.

More precisely, the invention concerns a device for analysing awavefront, of the Hartman or Shack-Hartmann type, comprising inparticular a set of sampling elements arranged in an analysis plane, andforming as many sub-pupils for sampling the incident wavefront, and adiffraction plane wherein are analysed the diffraction spots of thedifferent sub-pupils illuminated by the incident wavefront,characterised in that the shape of each sub-pupil is such that theassociated diffraction figure has in the diffraction plane one orseveral preferential axes, and in that the sub-pupils are oriented inthe analysis plane so that the preferential axes of the diffractionfigure of a sub-pupil are offset relative to the preferential axes ofthe diffraction figures of neighbouring sub-pupils, thereby enabling tolimit the overlapping of the diffraction figures.

According to a preferential embodiment, the sub-pupils are rectangularor square in shape, substantially identical, arranged in the form of atwo-dimensional matrix, and exhibit such an orientation relative to thedirections of the matrix that the preferential axes of the diffractionfigures of the sub-pupils in the diffraction plane are substantiallyparallel, not confused.

The invention enables to realise thus a wavefront analysis device, ofHartman type as well as of Shack-Hartmann type, whereof the resolutionis enhanced, while enabling to use sub-pupils used conventionally,without any specific apodisation components.

Other advantages and characteristics of the invention will appear moreclearly in the light of the following description, illustrated by theappended figures which represent:

on FIGS. 1A to 1D, round and square sub-pupil diagrams as well as theircorresponding diffraction figures,

on FIG. 2, the diagram of a matrix arrangement of rotated squaresub-pupils, according to an embodiment of the device according to theinvention,

on FIG. 3, a diagram of the diffraction figures issued from squaresub-pupils rotated by an angle enabling to reduce substantially theoverlapping phenomenon,

on FIGS. 4A and 4B, curves exhibiting the variation of the error inmeasuring the positions of the spots due to the overlapping phenomenonrelative to the displacement of the central spot, which for the 3 typesof sub-pupils studied: square, square rotated by an angle taken at 25°,round, for lateral and diagonal displacement,

on FIGS. 5A and 5B, error variation curves on the measurement of thepositions of the spots due to the overlapping phenomenon relative to thedisplacement of the central spot, for a matrix of rotated squaresub-pupils, namely for different spacing values between sub-pupils atconstant aperture.

An essential point of the wavefront analysis by the Hartmann orShack-Hartmann methods is the very accurate localisation in thediffraction plane of the positions of the focussing points issued fromthe sampling elements (micro-lenses or holes), when assuming priorcorrect sizing. Later on in the description, by wavefront analysis planeis meant the plane wherein are arranged the sampling elements anddiffraction plane, the plane of the diffraction spots, corresponding tothe diffraction plane when using micro-lenses.

FIGS. 1A to 1D illustrate the shapes of the sampling elements usedconventionally as well as the aspects of the corresponding diffractionfigures.

The Airy spot, noted FR, represented schematically on FIG. 1C,corresponds to the diffraction figure, in the diffraction plane, of theround sub-pupil noted R, represented schematically on FIG. 1A, andextends in an isotropic fashion.

The diffraction figure noted FC, represented schematically on FIG. 1D,corresponds to the diffraction figure, in the diffraction plane, of thesquare pupil noted C represented schematically on FIG. 1B and extendsalong 2 perpendicular directions, noted X and Y on the diagram,corresponding to both axes of the pupil.

The matrix arrangement of the elements described previously involves theoverlapping of the diffraction figures at the diffraction plane, sincefor the round pupil, the spot spread is isotropic and for the squarepupil, the matrix arrangement merges the axes of the neighbouringdiffraction figures.

Localising the spots, by calculating the gravity centre, is consequentlyrendered inaccurate, in particular during the displacement of one of thespots: a portion of the diffraction figure corresponding to one of thespots overlaps the neighbouring spot and involves fictitiousdisplacement of the gravity centres of said spots.

The device according to the invention enables maximum reduction of sucheffect.

According to the invention indeed, the shape of each sub-pupil is suchthat the associated diffraction figure exhibits in the diffraction planeone or several preferential axes, and the sub-pupils are oriented in theanalysis plane so that, the set of the sub-pupils being illuminated by aplane incident wavefront, the diffraction figure of a sub-pupil in saiddiffraction plane does not overlap substantially the diffraction figuresissued from the neighbouring sub-pupils. In other words, the sub-pupilsare oriented in the analysis plane so that the preferential axes of thediffraction figure of a sub-pupil are offset with respect to thepreferential axes of the diffraction figures of the neighbouringsub-pupils.

FIG. 2 illustrates an example of arrangement of the sub-pupils in adevice according to the invention. The sub-pupils noted C in the exampleof FIG. 2 are selected substantially identical, square in shape, whichis one of the shapes most commonly used in the Hartmann andShack-Hartmann type wavefront analysis devices. Such sub-pupils exhibitthe following advantages:

-   -   they are easy to produce,    -   the associated diffraction figures exhibit preferential        diffraction axes.

In the example of FIG. 2, the sub-pupils are moreover arranged in theform of a square matrix, matrix defined by these perpendicular axesX_(mat) and Y_(mat). In this example, each of the sub-pupils, noted FC,is rotated by angle θ relative to one of the axes of the matrix, whichwe shall consider as Xmat for the example, as representeddiagrammatically on FIG. 2.

FIG. 3 represents in the diffraction plane, the profile of thediffraction spots corresponding to the sub-pupils of FIG. 2. Thisarrangement enables, as represented diagrammatically on FIG. 3, tooffset the preferential spreading axes of the diffraction figures due toeach sub-pupil, and to minimise geometrically the overlappingphenomenon. Let us consider on the diagram of FIG. 3 the example of twodiffraction figures noted FC1 and FC2 issued from two neighbouringsub-pupils. The preferential spreading axes of these figures are notedrespectively X1, Y1 and X2, Y2. The rotation performed enables to offsetthe axes X1 and X2, Y1 and Y2 which were 2 by 2 confused in the case ofnon-rotated square sub-pupils. The angle of rotation noted θ of eachsub-pupil is selected so that the diffraction figures of twoneighbouring sub-pupils do not interfere substantially.

In order to quantify the effects of the overlapping phenomenon on thepositioning accuracy of the spots, we have simulated the diffraction ofa grid of micro-lenses, containing 5 by 5-sub-pupils arranged into asquare matrix, with a 670 nm wavelength, by addition in amplitudediffraction figures issued from each micro-lens taken separately. Thissimulation renders possible the selection of the shape of themicrolenses (square, round, rotated square), of their size, of theiraperture as defined hereunder, of the displacement of the image spotissued form a selected micro-lens, that we take here as being thecentral micro-lens of the matrix in order to quantify the effect of itsdisplacement over the largest zone possible. Simulation would be easilytransposable to the case of the Hartmann method since the diffractionfigures issued from micro-lenses or holes identical in shape aresubstantially equivalent.

A tilting angle of the sub-pupils is selected so that the overlappingphenomenon of the neighbouring diffraction figures is minimal. Thisangle depends on the distribution geometry of the sub-pupils: in thecase of a symmetrical geometry as is the case for our example, the valueof the angle is taken at 25°, enabling to offset at best thedistribution axes of the diffraction figures therebetween.

This simulation is carried out with a view to avoid any artefactassociated with the sampling: modelling operations are oversampled,while taking at least 100 samples per sub-pupil.

The aperture of the elementary analysis zone, i.e. the sub-pupil, is asignificant parameter of the system sizing. This parameter is defined asthe ratio of the focal distance of the micro-lenses to the size of thecorresponding sub-pupil. For the simulations performed, we consider a33-aperture, the best compromise between too wide an aperture whichwould consequently increase the size of the diffraction spots and hencethe overlapping between neighbouring spots, and too narrow an aperturewhich would consequently reduce excessively the size of the spots andrender too inaccurate a calculation of barycenter after integration by aCCD-type sensor.

Thresholding is set to 10% before calculation of the gravity centre ofeach spot, in order to simulate at best the thresholding carried outefficiently when detecting the spots by a CCD-type sensor, for instance,to break free from the detection noise.

We simulate the displacement of the image spot issued from the sub-pupilfrom the centre of this 5×5-matrix, by adding a tilting angle known onthe incident phase of the wavefront taken locally at this centralsub-pupil. The calculation of the barycenter on each of the spots issuedfrom the matrix enables us to trace the position of each of these spotsafter known displacement of the central spot. It is therefore possibleto calculate the error due to the overlapping phenomenon on the positionof each of the spots.

We select a displacement dynamic of the central spot corresponding to a20 μm maximum displacement, which corresponds, for the selected sizing,to a local phase variation at a sub-pupil of approximately thewavelength of the simulated beam, a significant variation and containingthe most part of the measurements made in wavefront analysis.

Finally, we simulate two types of displacements of the central spot:lateral displacement according to one of the two distribution axes ofthe sub-pupils on the matrix, diagonal displacement by equal andsimultaneous displacement according each of both axes. Taking intoaccount the distribution symmetry of the sub-pupils both these types ofdisplacement enable to give an accurate indication of the errorsgenerated, due to the overlapping phenomenon, by any type ofdisplacement in the dynamic selected above.

The calculation by barycenter of the position of the image spots afterdisplacement of the central spot enables to define a global qualitycriterion for the measuring accuracy of the positions of the spots, i.e.the variance, noted V, calculated from the errors on the position of themain spots by calculation of the barycenter after displacement of thecentral spot. The variance corresponds thus to the sum of the squares oferrors due to the overlapping phenomenon on the calculation of gravitycentre of the position of each spot. We shall note as main spots theneighbouring focalisation spots of the displaced spot which aresubjected to the largest influence of the overlapping phenomenon of thediffraction figures as regards the calculation of their position.

Rapid examination of the values of barycenter enables to identify thesemain spots as follows:

-   -   the central spot displaced    -   the four spots directly surrounding the central spot and        situated on either side thereof along both main axes of the        matrix.

FIG. 4 shows the evolution of the variance V defined above with thedisplacement of the central spot, namely for the three types of pupilconsidered: square, square rotate by a selected angle in order tominimise the overlapping phenomenon, round. FIG. 4A illustrates thisevolution relative to lateral displacement of the central spot, FIG. 4Brelative to a diagonal displacement of the same spot.

It can be noted thus that the geometrical arrangement performed enablesto improve the accuracy for calculating the positions of the spots withrespect to the shapes and dispositions of sub-pupils used hencetoforth,by a factor which may reach a 10-factor on the variance with respect toa matrix of non-rotated square sub-pupils.

A second simulation enables to visualise the improvement in resolutionof a Shack-Hartmann (or Hartmann) type device according to theinvention. To do so, a matrix of rotated square sub-pupils is consideredaccording to the method here developed, the spacing between thesub-pupils forming the analysis matrix is reduced, while modifying thefocal distance in order always to keep the same aperture.

FIG. 5 enables to quantify the evolution of the variance V defined aboverelative to the displacement of the central spot, for different valuesof the increase percentage in the resolution, noted G_(res), gaincorresponding to the reduction percentage of the spacing between thesub-pupils. FIG. 5A shows the results of this simulation for a lateraldisplacement, noted dx, of the central spot. FIG. 5B shows the resultsfor a diagonal displacement, noted dx=dy of the same spot.

We wish to quantify the increase in resolution that may be reachedthanks to the device according to the invention while keeping the sameperformances on the calculation of the position of the spots as a systemwhich would use a conventional matrix of sub-pupils.

The comparison of FIGS. 4A and 5A, 4B and 5B, enables to show that it ispossible to improve the resolution of the Hartmann or Shack-Hartmannwavefront analysis device while keeping the same 33%-measuring accuracyof the position of the spots with respect to a conventional matrix ofsquare sub-pupils, to 27% with respect to a matrix of round sub-pupils.

The embodiment example described previously is not limiting. Notably,other shapes of sub-pupils may be used in the device according to theinvention and other arrangements of the sub-pupils are possible as longas the diffraction figures corresponding to the sub-pupils exhibit oneor two preferential axes and as, in relation to the shapes selected, thesub-pupils are arranged in the analysis plane so that the preferentialaxes of the diffraction figures are offset relative to one another inorder to reduce the overlapping between the diffraction figures.

Thus, thanks to the device according to the invention, it is possible toincrease the number of micro-lenses or holes per surface unit withrespect to a matrix having a “conventional” geometrical arrangement ofthese micro-lenses or holes, while keeping the same accuracy forcalculating the position of the diffraction spots generated by thematrix. Thus, the number of sampling spots of the Shack-Hartmann orHartmann method is increased, and therefore its resolution.

1. A device for analysing a wavefront, of the Hartman or Shack-Hartmanntype, comprising in particular a set of sampling elements arranged in ananalysis plane, and forming as many sub-pupils (C) for sampling theincident wavefront, and a diffraction plane wherein the diffractionspots of the different sub-pupils illuminated by the incident wavefrontare analyzed, characterised in that the shape of each sub-pupil is suchthat the associated diffraction figure (FC) has in the diffraction planeone or several preferential axes (X1, Y1, X2, Y2), and in that thesub-pupils are oriented in the analysis plane so that the preferentialaxes of the diffraction figure of a sub-pupil are offset relative to thepreferential axes of the diffraction figures of neighbouring sub-pupils,thereby enabling to limit the overlapping of the diffraction figures,wherein the device is of the Hartman or Shack-Hartmann type.
 2. A deviceaccording to claim 1, characterised in that the sub-pupils are arrangedin the analysis plane in the form of a two-dimensional matrix, thesub-pupils are parallelepipedal in shape, the diffraction figure of asub-pupil having two preferential axes, and in that each sub-pupil isoriented with respect to the directions of the matrix so that thepreferential axes of the diffraction figure of said sub-pupil show anon-zero angle with the two axes of the two-dimensional matrix.
 3. Adevice according to claim 2, characterised in that the sub-pupils aresubstantially identical in shape, they show substantially the sameorientation with respect to the two axes of the two-dimensional matrixso that the preferential axes of the diffraction figures aresubstantially parallel, not confused.
 4. A device according to claim 2,characterised in that the sub-pupils are substantially rectangular inshape.
 5. A device according to claim 4, characterised in that thesub-pupils are substantially square in shape.
 6. A device according toclaim 1, characterised in that the sub-pupils are arranged in theanalysis plane in the form of a two-dimensional square matrix, in thatthe sub-pupils are square in shape, substantially identical, and in thateach sub-pupil is oriented with respect to the two axes of thetwo-dimensional matrix by an angle θ, this angle θ having a value thatis optimised to limit the overlapping of the diffraction figures.
 7. Adevice according to claim 1, characterised in that the sampling elementsare realised through apertures of predetermined shape, formed in anopaque screen.
 8. A device according to claim 1, characterised in thatthe sampling elements are realised by micro-lenses associated withapertures of predetermined shape.
 9. A device according to claim 2,characterised in that the sub-pupils are substantially rectangular inshape.
 10. A device for analysing a wave front, comprising: a set ofsampling elements arranged in an analysis plane and forming sub-pupils(C) for sampling an incident wavefront; and a diffraction plane whereindiffraction spots of different sub-pupils illuminated by the incidentwavefront are analyzed, wherein, the sub-pupils are arranged in theanalysis plane in the form of a two-dimensional matrix, each sub-pupilhas an associated diffraction figure (FC) with two preferential axes(X1, Y1, X2, Y2), the sub-pupils are oriented in the analysis plane sothat the preferential axes of the associated diffraction figures areoffset relative to the preferential axes of the diffraction figures ofneighboring sub-pupils, limiting overlapping of the diffraction figures,the device is of a Hartman or Shack-Hartmann type, and each sub-pupil isoriented so that the two preferential axes of the diffraction figure ofeach sub-pupil show a non-zero angle with the two axes of thetwo-dimensional matrix.
 11. A device according to claim 10, wherein, thesub-pupils are substantially identical in shape and show substantiallythe same orientation with respect to the two axes of the two-dimensionalmatrix.
 12. A device according to claim 10, wherein, the sub-pupils aresubstantially rectangular in shape.
 13. A device according to claim 10,wherein, the sub-pupils are substantially square in shape.
 14. A deviceaccording to claim 10, wherein, the sub-pupils are arranged in theanalysis plane in the form of a two-dimensional square matrix, and thesub-pupils are square in shape, substantially identical.
 15. A deviceaccording to claim 10, wherein, the sampling elements are realisedthrough apertures of predetermined shape, formed in an opaque screen.16. A device according to claim 10, wherein, the sampling elements arerealised by micro-lenses associated with apertures of predeterminedshape.
 17. A device according to claim 10, wherein, the sub-pupils areparallelepipedal in shape.
 18. A device for analysing a wavefront,comprising: sampling elements arranged in an analysis plane formingsub-pupils (C) for sampling an incident wavefront; a diffraction planewherein diffraction spots of different sub-pupils illuminated by theincident wavefront are analyzed, the sub-pupils arranged in the analysisplane in the form of a two-dimensional matrix, each sub-pupil having anassociated diffraction figure (FC) with two preferential axes (X1, Y1,X2, Y2), wherein, the sub-pupils are oriented in the analysis plane sothat the preferential axes of the associated diffraction figures areoffset parallel relative to the preferential axes of the diffractionfigures of neighboring sub-pupils, limiting the overlapping of thediffraction figures, and the device is one of a Hartman type and aShack-Hartmann type.